Related papers: On the normalized local volume of a non-closed poi…
This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of…
We introduce a notion of a point-wise entropy of measures (i.e local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…
We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow…
We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant $r$, $a$, $b$.
In this work, we find an explicit expression for the volume of the trace nonnegative polytope via a generalization of the Irwin-Hall distribution. This volume is an upper bound for the volume of all projected, normalized realizable spectra.…
Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and…
We derive a general closed expression for the local pressure exerted onto the corrugated walls of a channel confining a fluid medium. When the fluid medium is at equilibrium the local pressure is a functional of the shape of the walls. It…
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…
This is a companion to our previous paper. Here, we derive local dimension-free estimates for volumes of sub- and super-level sets of analytic functions of several variables.
The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free…
A formula of the renormalized volume of tubes over polalized K\"{a}hler-Einstein manifolds is given in terms of the Einstein constant and the volume of the polarization.
In this paper we study a notion of volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated…
We show the asymptotics of the volume density function in the class of central harmonic manifolds can be specified arbitrarily and do not determine the geometry.
We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a…
Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a klt singularity there exists a valuation with smallest normalized volume. We prove this conjecture and provide an example…
We study open sets $\mathcal{P}$ in normed spaces $X$ attaining a large volume while avoiding pairs of points at integral distance. The proposed task is to find sharp inequalities for the maximum possible $d$-dimensional volume. This…
The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…
Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.
The geometry of the dual amplituhedron is generally described in reference to a particular triangulation. A given triangulation manifests only certain aspects of the underlying space while obscuring others, therefore understanding this…